设实数x1,x2满足x1不等于 x2,a＞0,y1＝x1/(1+a) +ax2/(1+a) ,y2＝ax1/(1+a) +x2/(1+a)

题目

设实数x1,x2满足x1不等于 x2,a＞0,y1＝x1/(1+a) +ax2/(1+a) ,y2＝ax1/(1+a) +x2/(1+a)
则x1x2与y1y2的大小关系是什么

答案

设y1y2/x1x2=k,则
y1y2=kx1x2
(x1+ax2)(ax1+x2)=k(a+1)(a+1)x1x2
ax1x1+x1x2+aax1x2+ax2x2=k(x1x2+ax1x2+ax1x2+aax1x2)
由于x1x1+x2x2>=2x1x2
所以k>=1,y1y2>=x1x2